Averaged energy conditions and quantum inequalities

In this paper, connections are uncovered between the averaged weak (AWEC) and averaged null (ANEC) energy condtions, and quantum inequality restrictions (uncertainty-principle-type inequalities) on negative energy. In two- and four-dimensional Minkowski spacetime, we examine quantized, free massless, minimally coupled scalar fields. In a two-dimensional spatially compactified Minkowski universe, we derive a covariant quantum inequality-type bound on the difference of the expectation values of ${\mathit{T}}_{\mathrm{\mu }\nu }$ ${\mathit{u}}^{\mathrm{\mu }}$ ${\mathit{u}}^{\nu }$ in an arbitrary quantum state and in the Casimir vacuum state. From this bound, it is shown that the difference of expectation values also obeys AWEC- and ANEC-type integral conditions. This is surprising, since it is well known that the expectation value of ${\mathit{T}}_{\mathrm{\mu }\nu }$ ${\mathit{u}}^{\mathrm{\mu }}$ ${\mathit{u}}^{\nu }$ in the renormalized Casimir vacuum state alone satisfies neither quantum inequalities nor averaged energy conditions. Such ‘‘difference inequalities,’’ if suggestive of the general case, might represent limits on the degree of energy condition violation that is allowed over and above any violation due to negative energy densities in a background vacuum state. In our simple two-dimensional model, they provide physically interesting examples of new constraints on negative energy which hold even when the usual AWEC, ANEC, and quantum inequality restrictions fail. In the limit when the size of the space is allowed to go to infinity, we derive quantum inequalities for timelike and null geodesics which, in appropriate limits, reduce to AWEC and ANEC in ordinary two-dimensional Minskowski spacetime. Lastly, we also derive a covariant quantum inequality bound on the energy density seen by an arbitrary inertial observer in four-dimensional Minskowski spacetime. The bound implies that any inertial observer in flat spacetime cannot see an arbitrarily large negative energy density which lasts for an arbitrarily long period of time. From this latter bound, we derive AWEC and ANEC.